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## Section5.2Properties of Hermitian Matrices

The eigenvalues and eigenvectors of Hermitian matrices Section 5.1 have some special properties.

### The eigenvalues of a Hermitian matrix must be real.

To see why this relationship holds, start with the eigenvector equation

$$M |v\rangle = \lambda |v\rangle\tag{5.2.1}$$

and multiply on the left by $$\langle v|$$ (that is, by $$v^\dagger$$):

$$\langle v | M | v \rangle = \langle v | \lambda | v \rangle = \lambda \langle v | v\rangle\text{.}\tag{5.2.2}$$

But we can also compute the Hermitian conjugate (that is, the conjugate transpose) of (5.2.1), which is

$$\langle v| M^\dagger = \langle v| \lambda^*\text{.}\tag{5.2.3}$$

Using the fact that $$M^\dagger=M\text{,}$$ and multiplying by $$|v\rangle$$ on the right now yields

$$\langle v | M | v \rangle = \langle v | \lambda^* | v \rangle = \lambda^* \langle v | v \rangle\text{.}\tag{5.2.4}$$

Comparing (5.2.2) with (5.2.4) now shows that

$$\lambda^* = \lambda\tag{5.2.5}$$

as claimed.

### Eigenvectors corresponding to different eigenvalues must be orthogonal.

The argument establishing this relationship is similar to the one above. Suppose that

\begin{align} M |v\rangle \amp = \lambda |v\rangle ,\tag{5.2.6}\\ M |w\rangle \amp = \mu |w\rangle\text{.}\tag{5.2.7} \end{align}

Then

$$\langle v | \lambda | w \rangle = \langle v | M | w \rangle = \langle v | \mu | w \rangle\tag{5.2.8}$$

or equivalently

$$(\lambda - \mu) \langle v | w \rangle = 0\text{.}\tag{5.2.9}$$

Thus, if $$\lambda\ne\mu\text{,}$$ $$|v\rangle$$ must be orthogonal to $$|w\rangle\text{.}$$

In the case of repeated eigenvalues, it is possible to make a basis orthogonal using a process called Gram-Schmidt orthogonalization. (This amounts to simply subtracting off the parts of a given basis that are not orthogonal.)

### Conclusion:.

It is always possible to find an orthonormal basis of eigenvectors for any Hermitian matrix.

### Sensemaking5.1.Properties of Anti-Hermitian Matrices.

Repeat the two proofs above to find similar properties for anti-Hermitian matrices.

Hint.

Don't forget to use $$M^{\dagger}=-M\text{.}$$ Don't forget to complex conjugate the eigenvalue, where necessary.

Answer.

The eigenvalues are pure imaginary. The eigenvectors are still orthogonal.